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Observing a quantum Maxwell demon at workNathanael Cotteta,1, Sebastien Jezouina,1, Landry Bretheaua, Philippe Campagne-Ibarcqa, Quentin Ficheuxa,Janet Andersb, Alexia Auffevesc, Remi Azouitd,e, Pierre Rouchond,e, and Benjamin Huarda,f,2

aLaboratoire Pierre Aigrain, Ecole Normale Superieure, PSL Research University, CNRS, Universite Pierre et Marie Curie, Sorbonne Universites, UniversiteParis Diderot, Sorbonne Paris-Cite, 75231 Paris Cedex 05, France; bPhysics and Astronomy, College of Engineering, Mathematics, and Physical SciencesUniversity of Exeter, Exeter EX4 4QL, United Kingdom; cInstitut Neel, UPR2940 CNRS and Universite Grenoble Alpes, 38042 Grenoble, France; dCentreAutomatique et Systemes, Mines ParisTech, PSL Research University, 75272 Paris Cedex 6, France; eQuantic Team, INRIA Paris, 75012 Paris, France;and fLaboratoire de Physique, Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 7, France

Edited by Steven M. Girvin, Yale University, New Haven, CT, and approved June 5, 2017 (received for review March 23, 2017)

In apparent contradiction to the laws of thermodynamics,Maxwell’s demon is able to cyclically extract work from a sys-tem in contact with a thermal bath, exploiting the informationabout its microstate. The resolution of this paradox required theinsight that an intimate relationship exists between informationand thermodynamics. Here, we realize a Maxwell demon experi-ment that tracks the state of each constituent in both the classi-cal and quantum regimes. The demon is a microwave cavity thatencodes quantum information about a superconducting qubitand converts information into work by powering up a propa-gating microwave pulse by stimulated emission. Thanks to thehigh level of control of superconducting circuits, we directly mea-sure the extracted work and quantify the entropy remaining inthe demon’s memory. This experiment provides an enlighteningillustration of the interplay of thermodynamics with quantuminformation.

quantum thermodynamics | superconducting circuits |quantum information

In 1867, pondering the newly developed thermodynamic laws,Maxwell came to the disturbing conclusion that a “demon” can

extract work cyclically from a thermodynamic system beyond thelimits set by the second law when acting upon the information itobtains about the system (1). This paradox was resolved a cen-tury later when Landauer realized that information processinghas an entropic cost and Bennett argued that the demon’s mem-ory must take full part in the thermodynamic cycle (2). Recentexperiments have realized classical versions of elementaryMaxwell demons in various physical systems (3–8). Althoughquantum versions have long been investigated theoretically (9–13), experimental realizations are in their infancy (7, 8), and a fullcharacterization is still missing. Using superconducting circuits,we reveal the inner mechanics of a quantum Maxwell demon thatis able to extract work from a quantum system. Importantly, weare able to directly probe the extracted work by measuring theoutput power emitted by the system through stimulated emis-sion, without inferring it from system trajectories (3–6, 14). Weare thus able to demonstrate how the information stored in thedemon’s memory affects the extracted work. To make the charac-terization complete, we also measure the entropy and energy ofthe system and the demon. Superconducting circuits thus revealthemselves as a suitable experimental testbed for the bloomingfield of quantum thermodynamics of information (15–19).

In the experiment, the system S is a transmon supercon-ducting qubit (20) with energy difference hfS = h × 7.09 GHzbetween its ground |g〉 and excited |e〉 states. It is embedded ina microwave cavity that resonates at fD = 7.91 GHz and playsthe role of the demon’s memory D. The dispersive Hamiltonianreads H = hfS |e〉〈e|S + hfDd

†d − hχd†d |e〉〈e|S , where d is theannihilation operator of a photon in the cavity. The last terminduces a frequency shift of the cavity by −χ=−33 MHz whenthe qubit is excited. Reciprocally, the qubit frequency is shiftedby−Nχ when the cavity hosts N photons. This coupling enablescorrelation of the cavity with the qubit, by driving it through one

of the two microwave ports a and b. This correlation enablesthe extraction of work by the demon in an autonomous manner(Fig. 1).

Thermodynamic CycleWe now discuss the steps of the work extraction cycle. Duringstep 1, we prepare the system in a thermal state at an arbitrarytemperature TS ≥T 0

S (Fig. 1), where T 0S = 103± 9 mK is the

equilibrium system temperature in the dilution refrigerator. Thisthermalization is realized by driving, in a fraction p(TS ) of allexperimental sequences, the qubit with a resonant π-pulse, whichflips the qubit state, thus simulating thermalization with a heatbath (SI Appendix). Conveniently, this technique can also pre-pare nonthermal quantum states, such as an equal superpositionof the qubit, by driving it with a π/2-pulse.

Step 2 consists in encoding the state of the system into thedemon’s memory, which starts in the vacuum |0〉D . Driving porta with a pulse of amplitude αin at frequency fD (Fig. 1) excitesthe demon’s memory conditioned on the system being in |g〉S .This selectivity requires the pulse duration to be longer than χ−1

and shorter than the coherence times of the qubit and cavity (SIAppendix). By design, decoherence of both system and demon’smemory is dominated by spontaneous emission into port b withrespective relaxation rates γS = (2.2 µs)−1 and γD = (207 ns)−1.If the system starts in an arbitrary superposition cg |g〉S + ce |e〉S ,it becomes entangled with the demon (Fig. 1); ideally, cg |g〉S ⊗|α〉D + ce |e〉S ⊗ |0〉D , where |α〉D is a coherent state. In prac-tice, the qubit-induced nonlinearity and decoherence of the

Significance

Maxwell’s demon plays a central role in thermodynamics ofquantum information, yet a full experimental characterizationis still missing in the quantum regime. Here we use supercon-ducting circuits to realize a quantum Maxwell demon in whichall thermodynamic quantities can be controlled and measured.Using power detection resolved at the single microwave pho-ton level and unprecedented tomography techniques, wedirectly measure the extracted work while tracking the qubitand cavity entropies and energies. We are thus able to fullycharacterize the demon’s memory after the work extractionand show that it takes full part in the thermodynamic pro-cess. The experiment establishes superconducting circuits asa testbed well suited to perform quantum thermodynamicsexperiments.

Author contributions: N.C., S.J., L.B., P.C.-I., A.A., and B.H. designed research; N.C., S.J.,Q.F., and B.H. performed research; N.C., S.J., R.A., P.R., and B.H. analyzed data; and N.C.,S.J., L.B., J.A., and B.H. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1N.C. and S.J. contributed equally to this work.2To whom correspondence should be addressed. Email: benjamin.huard@ens-lyon.fr.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1704827114/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1704827114 PNAS | July 18, 2017 | vol. 114 | no. 29 | 7561–7564

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Fig. 1. Sketch of the quantum Maxwell demon experiment. (A) Afterpreparation (step 1) in a thermal or quantum state by a pulse at frequencyfS, the system S (superconducting qubit) state is recorded (step 2) into thedemon’s quantum memory D (microwave cavity). A pulse incoming towardport a at fD populates the cavity mode with a state ραin only if the qubit isin the ground state |g〉S. This information is used to extract work W (step3) which charges a battery B (a microwave pulse at frequency fS on port b)with one extra photon. Importantly, the system emits this photon only whenthe demon’s cavity is empty. The work is determined by amplifying and mea-suring the average output power at fS on bout . The memory reset (step 4)is performed by cavity relaxation. (B) When the system starts in a quantumsuperposition of |g〉S and |e〉S, the demon and system are entangled afterstep 2.

cavity lead to an impure memory state ραin instead of |α〉D . Theaverage photon number n = Tr(d†dραin ) is determined by fit-ting the numerical result of the full master equation to match theexperimentally obtained system state (SI Appendix).

The work extraction occurs during step 3. A coherentπ-pulse, playing the role of the battery B, is sent through portb at frequency fS (Fig. 1). Without the demon, the qubit woulddeterministically absorb (emit) a quantum of energy hfS from(into) the battery, if it is initially in |g〉S (|e〉S ). Crucially, thedemon prevents this transfer when its memory has N ≥ 1 pho-tons, because then the pulse is off resonance by −Nχ. When thecorrelation between the demon’s memory having no photons andthe system being in |e〉S is perfect, only stimulated emission isallowed, and work is extracted from system to battery. However,when the correlation is not perfect, in particular when n 1, thedemon sometimes erroneously lets the qubit absorb a quantumof energy from the battery.

The demon thus ends in a state with an entropy SD of at leastthe decrease of system entropy, and has to be reset to close thethermodynamic cycle (2). In the final step of this experiment(step 4), we let the demon’s memory thermalize with a secondbath that has a low temperature (72± 13 mK). Physically, thisbath differs from the one coupled to the system, because of thevarious operating frequency ranges of the microwave compo-nents on the output line and because of the likely coupling of thetransmon qubit to excited vibration modes of the substrate, whichcouple to the system more than to the demon. So this demon canextract work in a cyclic manner, but it does so using a secondbath, thus behaving as a regular heat engine.

Measuring the Extracted WorkRemarkably, the power extracted from the system during step3 when it is driven at fS can be directly accessed (SI Appendix)through the difference between the incoming and outgoing pho-ton rates of port b,

P

hfS= 〈b†outbout〉B − 〈b

†inbin〉B = γb

1 + 〈σZ 〉S2

+Ω

2〈σX 〉S ,

where σX , σY , and σZ are the Pauli matrices for the system;γb is the Purcell coupling rate of the system to the transmissionline through port b; and Ω ∝ |〈bin〉B | is the frequency of theRabi oscillations around σY induced by the drive. The two con-tributions on the far right side can be identified as spontaneousemission of the system through port b and stimulated emission.The latter is a coherent exchange of energy between the driveand system and, as such, contributes to the work extracted fromthe system.

In the experiment, at t = 0, we send a pulse bin with a durationπ/Ω with Ω = (67 ns)−1 γS ≥ γb so that spontaneous emis-sion can be neglected, and 〈b†outbout〉B − 〈b

†inbin〉B fully quan-

tifies the work extraction. We measure the field intensity onbout using a near-quantum-limited heterodyne detection setup(SI Appendix) to access directly the average instantaneous powerextracted from the system (21). The power is shown in Fig. 2 asa function of time during the pulse in step 3 in units of photonsper microsecond for various initial system states (Fig. 2, Inset)and for two values of the average photon number in the demonmemory n . In Fig. 2A, the average photon number n = 9 is largeenough for the demon to distinguish the system states well. Asexpected from the demon’s action, the measured power is pos-itive for all initial states and greater for higher initial system

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Fig. 2. Measured extracted power (normalized by a quantum of energyhfS) for step 3 as a function of time during the pulse at fS. Blue, orange,and red symbols and error bars correspond to an initial thermalized sys-tem at temperatures T = 0.17 K, 0.40 K, and above 8 K (see Inset for initialBloch vectors). Green symbols correspond to an initial quantum superposi-tion obtained by a 3π/2-pulse acting on the system at 0.10 K. Solid linesresult from a numerical simulation with no fit parameters and match themeasurements well. (A) The demon memory state ραin contains n = 9 pho-tons when encoding a system in |g〉S. (B) Same as A for an ignorant demon(n = 0 in step 2).

7562 | www.pnas.org/cgi/doi/10.1073/pnas.1704827114 Cottet et al.

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Fig. 3. Work and internal energy of the system. (A) Total extracted workduring step 3 as a function of

√n, with n the number of photons in the

demon memory. Symbols correspond to measurement of the battery, andsolid lines result from simulations. Colors correspond to the same initialstates of the system as in Fig. 2. (B) Symbols denote measured internalenergy US of the system at step 4 as a function of the amplitude αin ofthe pulse at fD used in step 2 to encode information in the demon’s memory.Error bars are smaller than symbol size. Solid lines result from the full masterequation and establish the conversion between αin and

√n. An additional

dark blue color corresponds to an initial temperature TS = T0S . Dashed lines

indicate US after preparation step 1. The equality of extracted work (A) andchange in US (B) is highlighted by dashed arrows of identical lengths.

temperature. In contrast, when the demon is unable to distin-guish |g〉S and |e〉S , which happens for αin = n = 0, the extractedpower is measured to be negative for the system starting in anythermal state (Fig. 2B). This counterproductive action occursbecause the demon is ignorant and lets the system drain energyfrom the battery. This failure uncovers the role of information inthe work extraction by the demon.

At n = 0, a distinctive feature appears in Fig. 2B when the sys-tem starts in a quantum superposition of |g〉S and |e〉S (green).Even though the total work is zero, just like for the equallymixed state (red), the instantaneous power now oscillates, illus-trating the work potential of coherences. In contrast, for an effi-cient demon (n = 9 in Fig. 2A), there is no quantum signaturein the extracted work. Note that the peak in the green curvearises due to overlapping of steps 2 and 3 to avoid transients(SI Appendix).

Integrating the extracted power over the duration of step 3gives the work W =

∫ π/Ω0

Pdt , whose magnitude is, at most,hfS . As αin increases, the demon’s encoding improves, andthe work increases from negative to positive values (Fig. 3A).This extracted work is given by the change in the systeminternal energy US = hfS 〈e|ρS |e〉 during step 3, W =−∆US −Q ≈−∆US , where Q is the heat arising from spontaneous emis-sion, which is negligible. Although the work was measured on

the battery, we independently measure US as a function of αin

(Fig. 3B) at the end of step 4 using the cavity as a dispersivedetector (20) (SI Appendix). The variations of work (Fig. 3A)indeed mirror the change of internal system energy (Fig. 3B)between steps 1 (dashed lines) and 4 (symbols). As αin increases,the demon extracts more energy from the system, making it endup close to the ground state (residual excitation of 2.7± 1%)whatever the initial state (Fig. 3B). Indeed, the thermodynamiccycle can be used to cool down superconducting qubits in prac-tice, as previously demonstrated in its continuous version (22).The full decay of US and the increase of W as a function of αin

are well reproduced numerically (solid lines in Fig. 3). It is nat-ural to compare the extracted work with Landauer’s work costof erasure, kBT ln 2 (2, 10). Because the system is connectedto a thermal bath only during step 1, the work extraction isnot optimal. Indeed, W is limited by the initial internal energyUS = hfS/[1 + exp(hfS/kBT )], which is at most 40% of the Lan-dauer bound.

Probing What the Demon RemembersA key signature of Maxwell’s demon is the transfer of entropyfrom the system to the memory (2). In contrast to previousrealizations of Maxwell demons (3–6), our experiment not onlyallows a direct measurement of work but also gives full accessto the density matrix ρD of the demon’s memory, includingits von Neumann entropy SD =−Tr(ρD ln ρD). We perform afull quantum tomography of ρD using the qubit as a measure-ment apparatus right after step 3 (SI Appendix) (23). When thequbit starts close to |g〉S , the maximum likelihood reconstruc-tion of the demon’s state gives ραin (SI Appendix) (24). When we

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Fig. 4. Tomography of the demon state. Reconstructed density matrixρD by maximum likelihood at the end of the work extraction step 4, forαin = 0.25 and when the system is initially (A) at temperature 0.10 K,(B) close to the excited state, (C) a superposition of ground and excitedstates, and (D) a maximally mixed state (see Bloch vector in Insets). Eachpixel represents the amplitude of a density matrix element in the Fock basis,and the von Neumann entropies SD are given. (Wigner function is shown inSI Appendix.)

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set αin = 0.25, the measured state ραin is found to be entropicand far from a coherent state, as expected (Fig. 4A). In con-trast, when the system starts in |e〉S , the memory stays closeto |0〉D , with a small residual entropy (Fig. 4B). Most inter-esting is the comparison of the effect on the demon when thesystem starts in a quantum superposition (Fig. 4C) versus in athermal state at large temperature (Fig. 4D). In the first case,SD = 1.0± 0.05, and ρD exhibits coherences between |0〉 andhigher Fock states, whereas coherences are missing in the sec-ond case, leading to a larger entropy SD = 1.2± 0.1. This trans-fer of nonclassicality from the system to the memory is a signa-ture of the quantum Maxwell demon. Although the entropiesof these two states are ordered as expected, their values aremuch larger than a bit of entropy, ln 2≈ 0.7. This excess entropyis quantitatively reproduced by simulations (SI Appendix) andarises because dissipation and nonlinearity of the memory resultsin encoding in a large number of energy levels rather than injust two dimensions. Using a full tomography of the system(SI Appendix), we have checked that the memory entropy SD

is always higher than the system entropy decrease SS (step 1)−SS (step 3).

Future developments of this experiment could involve super-conducting circuits with a widely tunable frequency, which wouldallow the implementation of optimal quasistatic processes, wherethe system stays in equilibrium. A test of Landauer’s principlecould then be realized in the quantum regime. The encodingfidelity of the demon is quantified by the mutual informationbetween system and demon. By adding an extra qubit and read-out cavity, one could demonstrate the expected proportionalitybetween extracted work and mutual information (25–27). Finally,with the level of control shown in the experiment, superconduct-ing circuits provide an exciting platform to explore single-shotquantum thermodynamics (28) and quantum heat engines (29).

ACKNOWLEDGMENTS. We thank M. Devoret, P. Degiovanni, E. Flurin,Z. Leghtas, F. Mallet, V. Manucharyan, J. Pekola, J. M. Raimond, M. Ueda,and the late M. Clusel for fruitful discussions and feedback. Nanofabrica-tion has been made within the consortium Salle Blanche Paris Center. Thiswork was supported by the Agence Nationale de la Recherche under Grants12-JCJC-TIQS and 13-JCJC-INCAL, by Ville de Paris through Grant Qumotelof the Emergence program, and by the COST network MP1209, “Thermo-dynamics in the quantum regime.” J.A. acknowledges support from Engi-neering and Physical Sciences Research Council, Grant EP/M009165/1, andthe Royal Society.

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